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#1 2007-12-12 17:02:58
- ganesh
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An old question
This is an old question for me. I have been asking this question from the time I was 15. That is, more than 23 years. I am posting the question with slight modifications.
Which of these is the biggest number?
1. The total number of grains of sand on planet earth.
2. The total number of leaves and leaflets on all the members of the Plant kingdom on planet earth.
3. The time taken by a snail to orbit around the galactic center once.
4. The mass of the sun, all the planets and their satellites in milligrams.
5. The value of 200! (That is 200x199x198x197x...3x2x1)
6. The smallest number apart from zero and one which is a perfect square, cube, fourth, fifth, sixth, ...and tenth power.
7. The probaility of choosing any two people from (roughly) the 7 billion people on planet earth.
Character is who you are when no one is looking.
#2 2007-12-13 02:57:04
- Ricky
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Re: An old question
3. The time taken by a snail to orbit around the galactic center once.
Unit of time not given, so I will represent time in 10^-k seconds, where k is the largest number from choices 1, 2, 4, 5, 6, and 7.
Therefore, the answer is 3.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
#3 2007-12-13 03:17:50
- NullRoot
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Re: An old question
Purely an academic question since the snail is using 10^-k seconds for his velocity, but was the orbit circular or elliptical and what was the min and max radius of his orbit?
Edit: Since it isn't stated, I declare the orbit to be 0 meters in circumference! The game's back on!
Last edited by NullRoot (2007-12-13 03:23:36)
Trillian: Five to one against and falling. Four to one against and falling… Three to one, two, one. Probability factor of one to one. We have normality. I repeat, we have normality. Anything you still can’t cope with is therefore your own problem.
#4 2007-12-13 03:30:20
- TheDude
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Re: An old question
Answers 1, 2, and 4 can be eliminated quickly. The mass of the sun is just less than 2*10^30 kg, which is 2*10^36 mg. Even if we dramatically overestimate and say that the rest of the solar system has 9 times the mass of the sun we'd only end up with 2*10^37 mg of mass for number 4, which is quite small. Numbers 1 and 2 will be much smaller yet than this number, eliminating them.
Number 7 is also quite a small number. It's just 7 billion squared, which is about 4.9*10^19, again much too small for this question.
When we compare 5 and 6, it's simple to show that 6 is much larger than 5. It's easy to show that 200! is much less than 200^200 (200! = 200*199*198*..., while 200^200 = 200*200*200*..., so each term is larger than it's factorial counterpart). We know that 200^200 = 2^200 * 100^200.
Now, consider number 6. The smallest such number will be 2 to a power that can be divided by 2, 3, 4, ..., 10. The smallest such number can be found by factoring each number into it's primes and finding the smallest such number that includes every prime factor. This number is 2^3 * 3^2 * 5^1 * 7^1 = 2520. Thus, number 6 is 2^2520 = 2^200 * 2^2320. All you must do now is convince yourself that 2^2320 is greater than 100^200. Consider that 2^7 = 128 > 100, so we have 2^2320 > (2^7)^330 = 128^330 > 100^200. Thus, we know that number 6 is a larger number than 1, 2, 4, 5, and 7.
That leaves only number 3. As Ricky pointed out there is no time unit given. The sun (and therefore Earth) take about 2.5*10^8 years to travel around the galaxy. This is roughly 8*10^15 seconds. Rounding down, the plank time is 10^-45 seconds. That means that even if we use the smallest unit of time that has any physical significance it would only take a snail 8*10^60 time units to circle the galaxy, far less than 2^2520. Thus, I declare number 6 the winner.
Wrap it in bacon
#5 2007-12-13 16:11:19
- ganesh
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Re: An old question
Wonderful work, Ricky, NullRoot and TheDude.
I thank NullRoot and TheDude for coming to my rescue! I had missed to mention seconds as the unit for #3.
The answer given by TheDude is perfectly in order. The number is roughly 3.9 x 10^758. Commendable work, TheDude!
Character is who you are when no one is looking.
#8 2008-10-29 22:56:39
- TheDude
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Re: An old question
The question reads "The smallest number apart from zero and one which ...".
Wrap it in bacon